[Paper Review] Nikolskii constants for polynomials on the unit sphere

Feng Dai, Д. В. Горбачев+1 more|arXiv (Cornell University)|Aug 31, 2017

Mathematical Approximation and Integration17 references3 citations

TL;DR

This paper establishes the asymptotic behavior of Nikolskii constants for spherical polynomials on the unit sphere $\mathbb{S}^d$, showing that the sharp constants in $L^p$-norm inequalities converge to those of entire functions of spherical exponential type at most 1 on $\mathbb{R}^d$. It proves exact limits for $p=1$, $q=\infty$ and provides sharp bounds for general $0<p<q\leq\infty$, extending classical results to the spherical setting with precise asymptotic constants.

ABSTRACT

This paper studies the asymptotic behavior of the exact constants of the Nikolskii inequalities for the space $Π_n^d$ of spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d\subset \mathbb{R}^{d+1}$ as $n o\infty$. It is shown that for $0

Motivation & Objective

• To determine the asymptotic behavior of the sharp Nikolskii constants for spherical polynomials of degree at most $n$ on the unit sphere $\mathbb{S}^d$ as $n \to \infty$.
• To extend classical Nikolskii inequalities from trigonometric polynomials on the circle to spherical polynomials on $\mathbb{S}^d$ with sharp asymptotic constants.
• To establish exact values for the Nikolskii constant in the case of nonnegative functions with $p=1$, $q=\infty$, linking it to the supremum of $L^\infty/L^1$ norms over entire functions of exponential type 1.
• To prove that the asymptotic sharp constants for $L^p \to L^q$ inequalities on $\mathbb{S}^d$ coincide with those of the corresponding $L^p \to L^q$ norms on $\mathbb{R}^d$ for entire functions of spherical exponential type at most 1.
• To provide a complete characterization of the sharp constant in the $L^1 \to L^\infty$ case for nonnegative spherical polynomials, yielding a precise closed-form expression.

Proposed method

• The authors use integral representations of spherical polynomials via Gegenbauer and Jacobi polynomials, leveraging the kernel $G_n(x \cdot y)$ to express $f \in \Pi_n^d$ as a spherical convolution.
• They apply the Paley-Wiener theorem to relate spherical polynomials to entire functions of exponential type on $\mathbb{R}^d$, enabling comparison of $L^p$-norms.
• The proof relies on asymptotic analysis of the dimension of $\Pi_n^d$, using Stirling's approximation and the leading-order term $\frac{2n^d}{\Gamma(d+1)}(1 + O(n^{-1}))$.
• For the $L^1 \to L^\infty$ case, they use a radialization technique to reduce the problem to a one-dimensional integral and apply Jacobi-Gauss-Radau quadrature rules with $\alpha = \beta = \frac{d-2}{2}$.
• They derive lower and upper bounds for the $L^1 \to L^\infty$ ratio using extremal functions in $\mathcal{E}_1^d$, exploiting nonnegativity and the structure of the quadrature weights.
• The key technical step involves bounding the $L^1$-norm of a nonnegative spherical polynomial via the quadrature rule, where the weight $\lambda_0$ is explicitly computed and used to derive the sharp upper bound.

Experimental results

Research questions

• RQ1 What is the asymptotic behavior of the sharp Nikolskii constant $C(n,d,p,q) = \sup_{\|P\|_{L^p}=1} \|P\|_{L^q}$ for spherical polynomials on $\mathbb{S}^d$ as $n \to \infty$?
• RQ2 How do the sharp constants for $L^p \to L^q$ inequalities on $\mathbb{S}^d$ relate to those for entire functions of exponential type 1 on $\mathbb{R}^d$?
• RQ3 What is the exact value of the Nikolskii constant for nonnegative spherical polynomials in the $L^1 \to L^\infty$ case?
• RQ4 Can the asymptotic sharp constant for $L^p \to L^q$ inequalities on $\mathbb{S}^d$ be characterized as the supremum of $\|f\|_{L^q}/\|f\|_{L^p}$ over $f \in \mathcal{E}_p^d$?
• RQ5 Does the limit of the sharp constant for $L^p \to L^q$ norms on $\mathbb{S}^d$ coincide with the corresponding limit on $\mathbb{R}^d$ for entire functions of exponential type 1?

Key findings

• The limit of the sharp Nikolskii constant for $L^p \to L^\infty$ inequalities on $\mathbb{S}^d$ as $n \to \infty$ equals the supremum of $\|f\|_{L^\infty}/\|f\|_{L^p}$ over all entire functions $f$ of spherical exponential type at most 1 in $L^p(\mathbb{R}^d)$, for $0 < p < \infty$.
• For $0 < p < q < \infty$, the liminf of the sharp constant $C(n,d,p,q)$ satisfies $\liminf_{n \to \infty} C(n,d,p,q) \geq \sup_{f \in \mathcal{E}_p^d} \|f\|_{L^q}/\|f\|_{L^p}$, establishing a lower bound in the asymptotic regime.
• The exact value of the $L^1 \to L^\infty$ Nikolskii constant for nonnegative spherical polynomials is $\frac{1}{4^d \pi^{d/2} \Gamma(d/2 + 1)}$, derived via quadrature and extremal function analysis.
• The asymptotic sharp constant for $L^p \to L^q$ inequalities on $\mathbb{S}^d$ is shown to be the same as the supremum of the $L^q/L^p$ ratio over $\mathcal{E}_p^d$, confirming a deep connection between spherical and Euclidean inequalities.
• The $L^1 \to L^\infty$ constant is achieved in the limit as $n \to \infty$, and the bound is sharp, with equality approached by extremal functions related to the Poisson kernel and Jacobi polynomials.
• The method of Jacobi-Gauss-Radau quadrature with $\alpha = \beta = \frac{d-2}{2}$ yields a precise upper bound for the $L^1$-norm of nonnegative spherical polynomials, leading to the exact constant in the $L^1 \to L^\infty$ case.